Abstract
Computing the convex envelope is a core operation in nonsmooth analysis that bridges the convex with the nonconvex world. Although efficient algorithms to compute fundamental transforms of convex analysis have been proposed over the years, they are limited to convex functions until an efficient algorithm becomes available to compute the convex envelope of a piecewise linear-quadratic function (of one variable) efficiently. We present two such algorithms, one based on maximum and conjugate computation that is easy to implement but has quadratic time complexity, and another based on direct computation that requires more work to implement but has optimal (linear time) complexity. We prove their time (and space) complexity, and compare their performances.
Similar content being viewed by others
References
Bauschke, H.H., Goebel, R., Lucet, Y., Wang, X.: The proximal average: basic theory. SIAM J. Optim. 19, 768–785 (2008)
Bauschke, H.H., Lucet, Y., Trienis, M.: How to transform one convex function continuously into another. SIAM Rev. 50, 115–132 (2008)
Bauschke, H.H., Wang, X.: The kernel average of two convex functions and its application to the extension and representation of monotone operators. Trans. Am. Math. Soc. 361, 5947–5965 (2009)
Bremner, D., Chan, T.M., Demaine, E.D., Erickson, J., Hurtado, F., Iacono, J., Langerman, S., Taslakian, P.: Necklaces, convolutions, and X + Y. In: Algorithms—ESA 2006. Lecture Notes in Computer Science, vol. 4168, pp. 160–171. Springer, Berlin (2006)
Brenier, Y.: Un algorithme rapide pour le calcul de transformées de Legendre–Fenchel discrètes. C. R. Acad. Sci. Paris Sér. I Math. 308, 587–589 (1989)
Corrias, L.: Fast Legendre–Fenchel transform and applications to Hamilton–Jacobi equations and conservation laws. SIAM J. Numer. Anal. 33, 1534–1558 (1996)
de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Algorithms and applications. In: Computational Geometry, 3rd edn. Springer, Berlin (2008)
Edelsbrunner, H.: Algorithms in Combinatorial Geometry. EATC Monographs on Theoretical Computer Science. Springer (1987)
Felzenszwalb, P.F., Huttenlocher, D.P.: Distance Transforms of Sampled Functions. Tech. Rep. TR2004-1963, Cornell Computing and Information Science (2004)
Gardiner, B., Lucet, Y.: Graph-Matrix Calculus for Computational Convex Analysis. Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Verlag series Optimization and Its Application (2010, in press)
Hare, W.: A proximal average for nonconvex functions: a proximal stability perspective. SIAM J. Optim. 20, 650–666 (2009)
Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex analysis and minimization algorithms. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vols. 305–306. Springer, Berlin (1993); Vol I: Fundamentals. Vol II: Advanced Theory and Bundle Methods
Hiriart-Urruty, J.-B., Lucet, Y.: Parametric computation of the Legendre–Fenchel conjugate. J. Convex Anal. 14, 657–666 (2007)
Koch, V., Johnstone, J., Lucet, Y.: Convexity of the proximal average. J. Optim. Theory Appl. (2010, in press)
Lachand-Robert, T., Oudet, É.: Minimizing within convex bodies using a convex hull method. SIAM J. Optim. 16, 368–379 (2005, electronic)
Laraki, R., Lasserre, J.B.: Computing uniform convex approximations for convex envelopes and convex hulls. J. Convex Anal. 15, 635–654 (2008)
Lucet, Y.: A fast computational algorithm for the Legendre–Fenchel transform. Comput. Optim. Appl. 6, 27–57 (1996)
Lucet, Y.: Faster than the fast Legendre transform. The linear-time Legendre transform. Numer. Algorithms 16, 171–185 (1997)
Lucet, Y.: A linear Euclidean distance transform algorithm based on the linear-time Legendre transform. In: Proceedings of the Second Canadian Conference on Computer and Robot Vision (CRV 2005), pp. 262–267, Victoria BC. IEEE Computer Society Press (2005)
Lucet, Y.: Fast Moreau envelope computation I: numerical algorithms. Numer. Algorithms 43, 235–249 (2006)
Lucet, Y.: What shape is your conjugate? A survey of computational convex analysis and its applications. SIAM J. Optim. 20, 216–250 (2009)
Lucet, Y., Bauschke, H.H., Trienis, M.: The piecewise linear-quadratic model for computational convex analysis. Comput. Optim. Appl. 43, 95–118 (2009)
Moreau, J.-J.: Proximité et dualité dans un espace Hilbertien. Bull. Soc. Math. Fr. 93, 273–299 (1965)
Noullez, A., Vergassola, M.: A fast Legendre transform algorithm and applications to the adhesion model. J. Sci. Comput. 9, 259–281 (1994)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)
She, Z.-S., Aurell, E., Frisch, U.: The inviscid Burgers equation with initial data of Brownian type. Commun. Math. Phys. 148, 623–641 (1992)
Trienis, M.: Computational Convex Analysis: From Continuous Deformation to Finite Convex Integration. Master’s Thesis, University of British Columbia (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gardiner, B., Lucet, Y. Convex Hull Algorithms for Piecewise Linear-Quadratic Functions in Computational Convex Analysis. Set-Valued Anal 18, 467–482 (2010). https://doi.org/10.1007/s11228-010-0157-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11228-010-0157-5
Keywords
- Computer-aided convex analysis
- Computational convex analysis
- Piecewise linear-quadratic functions
- Quadratic spline
- Moreau envelope
- Legendre–Fenchel conjugate